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- W4238486543 abstract "Let ${z_{nk}} = {e^{i{t_{nk}}}}$, $0 leq {t_{n0}} < cdots < {t_{nn}} < 2pi$, $f$ a function in the disc algebra $A$, and ${mu _n} = max { |{t_{nk}} - 2kpi /(n + 1)|:0 leq k leq n}$. Denote by ${L_n}(f;; cdot )$ the polynomial of degree $n$ that agrees with $f$ at ${ {z_{nk}}:k = 0, ldots ,n}$ . In this paper, we prove that for every $p$, $0 < p < infty$, there exists a ${delta _p} > 0$, such that $||{L_n}(f;cdot ) - f|{|_p} = O(omega (f;frac {1} {n}))$ whenever ${mu _n} leq {delta _p}/(n + 1)$. It must be emphasized that ${delta _p}$ necessarily depends on $p$, in the sense that there exists a family ${ {z_{nk}}:k = 0, ldots ,n}$ with ${mu _n} = {delta _2}/(n + 1)$ and such that $||{L_n}(f;cdot ) - f|{|_2} = O(omega (f;frac {1} {n}))$ for all $f in A$, but $sup { ||{L_n}(f;cdot )|{|_p}:f in A,||f|{|_infty } = 1}$ diverges for sufficiently large values of $p$. In establishing our estimates, we also derive a Marcinkiewicz-Zygmund type inequality for ${ {z_{nk}}}$." @default.
- W4238486543 created "2022-05-12" @default.
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- W4238486543 date "1993-04-01" @default.
- W4238486543 modified "2023-09-24" @default.
- W4238486543 title "On Lagrange Interpolation at Disturbed Roots of Unity" @default.
- W4238486543 doi "https://doi.org/10.2307/2154377" @default.
- W4238486543 hasPublicationYear "1993" @default.
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